APPLIED PROBABILITY AND STOCHASTIC PROCESSES

INTERNATIONAL SERIES IN OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Frederick S. Hillier, Series Editor Stanford University

Saigal, R: LINEAR PROGRAMMING: A MODERN INTEGRATED ANALYSIS

Nagurney, A. & Zhang, D.: PROJECTED DYNAMICAL SYSTEMS AND VARIATIONAL INEQUALITIES WITH APPLICATIONS

Padberg, M. & Rijal, M.: LOCATION, SCHEDULING, DESIGN AND INTEGER PROGRAMMING

Vanderbei, R: LINEAR PROGRAMMING: FOUNDATIONS AND EXTENSIONS

Jaiswal, N.K.: MILITARY OPERATIONS RESEARCH: QUANTITATIVE DECISION MAKING

Gal, T. & Greenberg, H.: ADVANCES IN SENSITIVITY ANALYSIS AND PARAMETRIC PROGRAMMING

Prabhu, N.U: FOUNDATIONS OF QUEUEING THEORY Fang, S.-c., Rajasekera, IR & Tsao, H.-S.I: ENTROPY OPTIMIZATION

AND MATHEMATICAL PROGRAMMING Yu, G.: OPERATIONS RESEARCH IN THE AIRLINE INDUSTRY Ro, T.-H. & Tang, C.S.: PRODUCT VARIETY MANAGEMENT EI-Taha, M. & Stidham, S.: SAMPLE-PATH ANALYSIS OF QUEUEING

SYSTEMS Miettinen, K.M.: NONLINEAR MULTI OBJECTIVE OPTIMIZATION Chao, H. & Huntington, H.G.: DESIGNING COMPETITIVE ELECTRICITY

MARKETS Weglarz, 1: PROJECT SCHEDULING: RECENT MODELS, ALGORITHMS &

APPLICATIONS Sahin, I. & Polatoglu, H.: QUALITY, WARRANTY AND PREVENTIVE

MAINTENANCE Tavares, L.v.: ADVANCED MODELS FOR PROJECT MANAGEMENT Tayur, S., Ganeshan, R & Magazine, M.: QUANTITATIVE MODELING FOR

SUPPLY CHAIN MANAGEMENT Weyant, 1: ENERGY AND ENVIRONMENTAL POLICY MODELING Shanthikumar, IG. & Sumita, U: APPLIED PROBABILITY AND

STOCHASTIC PROCESSES

APPLIED PROBABILITY AND STOCHASTIC PROCESSES

edited by

J. G. Shanthikumar and Ushio Sumita

~.

" Springer Science+Business Media, LLC

Library of Congress Cataloging-in-Publication Data

Applied probability and stochastic processes / edited by 1. G. Shanthikumar and Ushio Sumita.

p. cm. - (International series in operations research & management science: 19)

Includes bibliographical references and index. ISBN 978-1-4613-7364-3 ISBN 978-1-4615-5191-1 (eBook) DOI 10.1007/978-1-4615-5191-1 1. Probabilities. 2. Stochastic processes. I. Shanthikumar,1. George.

11. Sumita, U. (Ushio), 1949- . III. Series. QA273.18.A55 1999 519.5-dc21 99-17291

Copyright © 1999 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers, New York in 1999 Softcover reprint of the hardcover 1 st edition 1999

eIP

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC.

Printed on acid-free paper.

CONTENTS

Tribute to Julian Keilson VII

List of Publications by Julian Keilson xv

Chapter 1 Comments on the Perturbation Method 1 R. Syski and N. Liu

Chapter 2 Some Aspects of Complete Monotonicity in Time-Reversible Markov Chains 17 Mark Brown

Chapter 3 Transformations of Poisson Processes: Particle Systems and Networks 25 Richard F. Serfozo

Chapter 4 On the Local Time of the Brownian Bridge Lajos Takacs

Chapter 5 Probabilistic Token Causation: A Bayesian Perspective Elja Arjas

Chapter 6 On a Statistical Algorithm to Decode Heavily Corrupted Linear Codes I. N. Kovalenko and M. N. Savchuk

Chapter 7 Mean Cover Times for Coupon Collectors and Star Graphs Erol Pekoz and Sheldon M. Ross

Chapter 8 Models for the Spread of Infection via Pairing at Parties D. 1. Daley and 1. Gani

45

63

73

83

95

vi

Chapter 9 Extremes of Random Numbers of Random Variables: A Survey 115 Moshe Shaked and Tityik Wong

Chapter 10 Optimality of Sequential Quality Control via Stochastic Orders 129 David D. Yao and Shaohui Zheng

Chapter 11 Reallocatable GSMP with Sequentially Dependent Lifetimes: Clockwise Decomposability and Its Applications 149 Masakiyo Miyazawa

Chapter 12 Random Matrices and the Number of to, I} Matrices with Given Rowand Column Sums 173 Teunis J. Ott and J. George Shanthikumar

Chapter 13 Monotone Optimal Policies for Left-Skip-Free Markov Decision Processes 191 Shaler Stidham Jr. and Richard R. Weber

Chapter 14 Optimal Routing Control in Retrial Queues 203 H. M. Liang and V. G. Kulkarni

Chapter 15 Waiting TImes when Service TImes are Stable Laws: Tamed and Wild 219 Donald P. Gaver and Patricia A. Jacobs

Chapter 16 Winning the Hand of the Princess Saralinda 231 Peter W. Glynn and Ward Whitt

Chapter 17 Analysis of Multiple Queues with Passing Servers 247 Ushio Sumita, Masaaki Sibuya, and Norihiko Miyawaki

Chapter 18 Some Properties of Throughput in a Queueing Network with Changing-Speed Servers and Blocking 263 Genji Yamazaki and Hirotaka Sakasegawa

Chapter 19 Quasi-stationary Distributions of Markov Chains Arising from Queueing Processes: A survey 277 Masaaki Kijima and Naoki Makimoto

Chapter 20 Estimating Customer Loss Rates From Transactional Data 313 D. J. Daley and L. D. Servi

Index 333

TRIBUTE TO JULIAN KEILSON

J. G. Shanthikumar and Ushio Sumita

More than 45 years have passed since Professor Keilson began contributing to prob­ability theory. His intellectual roots are in mathematics and physics. At Brooklyn College's mathematics department, which nurtured such substantial mathemati­cians as Richard Bellman, he fell in love with mathematical analysis and with the beauty of nineteenth-century potential theory. In the belief that theoretical physics would provide a home for his mathematical needs, he switched his major to physics in his senior year and entered Harvard's doctoral program in the summer of 1947. Further strength in mathematical physics developed under Julian Schwinger, from whom he received his Ph.D. in June of 1950. Since then, starting first as a theoreti­cal physicist, he has been making tremendous contributions in various areas, includ­ing applied probability, stochastic processes, and computational probability, among others. In what follows, his major contributions are briefly summarized by classify­ing them into several categories. Paper references can be found in the list of publi­cations by Professor Keilson that follows this summary.

From Physics to Probability

Discomfort with the formal, heuristic, and pragmatic character of quantum mechan­ics and quantum electrodynamics led him back to mathematics. As a postdoctoral Fellow in Electronics at Harvard (1950--1952), he became interested in Brownian motion and the theory of electrical noise. His first paper [1] with J. Storer addressed the bridge between the jump character of particle motion sample paths described by Boltzmann's integral equation and the continuous sample path character of the Fokker-Planck partial differential equation. Related papers [2-5] at M.I.T.'s Lincoln Laboratory (1952-1956) studying diffusion in semiconductors were tied to transistor technology, then in its infancy. Just as Professor Philip Morse's role in early aspects of operations research was based on his strength in mathematical physics, these diffusion and noise studies became the foundation for Professor Keilson's subsequent activity in applied probability.

viii

State-Space Analysis

A multivariate Markov process may often be analyzed by studying changes with time of its state-space distribution. For a physicist steeped in multivariate Markov diffusion processes, such an approach was natural. Two influential papers with A. Kooharian [6,8] treated the transient behavior of the occupancy process of M/G/1 in this way. A similar procedure was introduced simultaneously by David Wishart. Even though direct probabilistic arguments are sometimes easier, such argumenta­tion can be very tricky and convoluted. The state-space approach has the advantage of being systematic and less prone to error. Indeed, the state space serves as a huge tableau on which the probabilistic motion is depicted.

Green's Function Methods

This activity was initiated at Sylvania's Applied Research Laboratory (1956-1966), then a part of the Sylvania Electronic Systems Division at Waltham. Some of Professor Keilson's first papers in queueing theory [7,9] attacked the difficult occu­pancy process N(t) of an MIMll system with time-dependent arrival rate l(t) and service rate m(t) by employing the Green's function methods of mathematical physics. The basic idea was simple. If there were no boundary at zero, the process would be spatially homogeneous and easy to treat. The trick needed was to act as if the process were on the lattice of all integers ( ... -2,-1,0,1,2 ... J and to create the effect of the boundary at zero by injecting signed mass density at states ° and -1. A treatment of the transient behavior of MIMls was available by the same method

A Markov process X(k + 1) = max[O, X(k) + x(k + 1)], where x(k) are Li.d. incre­ments, is encountered in the Lindley process and in queues with bulk arrivals and bulk service. Analysis of such processes is normally treated by Wiener-Hopf or Hilbert problem factorization methods in the complex plane where human intuition founders. When such processes are regarded as additive spatially homogeneous processes modified by a boundary at zero, Green's function methods and injection of compensating sources are again relevant. The analysis is conducted entirely in the real domain and quick answers are obtained intuitively. For example, the Pollaczek-Khinchine waiting time distribution for an M/G/1 system pops out in a few lines. A series of papers on these topics appeared [10,11,14,18-20], and multi­variate extensions and ties to M/G/1 were presented. A more extensive monograph devoted to problems in one dimension was written for the Charles Griffin series (see Monograph 1 at the end of the references). There, unfortunately, the real­domain character of the method was masked by characteristic function notation in the complex domain. Had moment-generation function notation been used instead, the method would have been seen to be user friendly and its real-domain charac­ter apparent. The fact that the Pollaczek-Khinchine waiting time distribution emerges in a few lines with real-domain argumentation is striking.

This real-domain method permits structural insights that are otherwise concealed. The loss formula for MIG/1 systems with finite capacity was obtained quickly in this way. Subsequent extension of the loss formula to more complex settings was demon­strated with L. Servi [117].

ix

Additive Processes Defined on a Finite Markov Chain

In the spring of 1963, a long collaboration was initiated with David Wishart, then at the University of Birmingham, England. Concern centered about the Central Limit Theorem for sums of increments dependent on an underlying finite Markov chain J(k). Let bmn be the transition probability for m to n and let Xmn be the incre­ment for m to n. The matrix b (s) = [bmnE[exp(sxmn)] is then the matrix generating function for the increments. Its behavior for s imaginary near s = 0 is the basis for a Central Limit Theorem as in the univariate case. Of interest was the quantifica­tion of the asymptotic growth rate of the mean and the variance, and these were obtained in terms of the first two derivatives at s = 0 of the principal eigenvalue of b (s) [21,23]. The two growth rates were subsequently exhibited explicitly, both probabilistically and analytically. The matrix b (s) and its principal eigenvalue are of similar interest to Markov renewal theory [26]. Similar results were obtained with S. Subba Rao for the additive process S(t) in continuous time for which

:t SU) = J(J(t», where J(t) is an underlying finite Markov chain in continuous time

137,38].

Interrupted Service

A simple formula for the effective service time distribution when service is inter­rupted preemptively at Poisson epochs with i.i.d. delays before resumption of service was exhibited [13]. This result was also found simultaneously by D. Gaver. The result is useful in many contexts. It unlocks queuing systems with customer classes of different preemptive priority.

Strong Unimodality for Lattice Distributions

Let a p.d.f. a(x) be strongly unimodal, i.e., let a(x) be unimodal and let b(x) con­volved with a(x) be unimodal whenever b(x) is unimodal. It had been shown by Ibragimov that a necessary and sufficient condition for such strong unimodality is that log a(x) be concave on the interval of support. It was shown by Keilson with H. Gerber [41] that a distribution on the lattice of integers with masses Pn is strongly unimodal in the same sense if and only if p/ ~ Pn+l Pn-l for all n. Most classical lattice distributions are found to have this property.

Measures of Exponentiality and Normality

In a useful paper with EW. Steutel [52], it was shown that for the set of completely monotone c.d.f.s, a simple distance to pure exponentiality in a metric-space sense is

2

provided by _s_ -1, the excess of the squared coefficient of variation over l.1t was m2

also shown that in the rich space of p.d.f.sJ(x) = g(x) *g(-x), where g(x) is a mixture

x

of symmetric normal p.d.f.s, the kurtosis is a measure of departure from normality in a metric-space sense. Error bounds for the departure from pure exponentiality of a completely monotone p.d.f. were established subsequently by Mark Brown. Error bounds for normality were given by C. Heidi.

Unimodality of Passage TImes in Birth-Death and Other TIme-Reversible Processes

It was shown in [42] that, for any birth-death process, the first-passage time density from the lowest-state zero to any state n is a sum of independent exponential vari­ates and is hence strongly unimodal and log concave. In the same paper, it was demonstrated that the sojourn time or dwell time on any set tn: n ~ no) is com­pletely monotone and hence log convex. This structure of sojourn time densities was extended subsequently [47] to any time-reversible chain. In a separate paper [76], it was shown some years later that for any birth-death process for which the passage time densities smn(t) from state m to state n exist, all such densities are unimodal. Indeed, any such passage time density is the convolution of two densities, one log concave and the other log convex.

Rarity and Exponentiality

In many stochastic systems having a regenerative state, one is often interested in the behavior of a passage time density to some subset A of the state space of inter­est, which is rarely visited. As a simple example, one might consider the time to sat­uration of an MIM/K/K system. Insofar as the passage time from the regenerative state can be regarded as a geometrically distributed number of independent inter­vals between regenerations followed by a final arrival to A without having regen­erated, one would guess that the passage time is approximately exponential in distribution. This exponentiality was established within the framework of a limit theorem [27] of practical and aesthetic interest. The regenerative property is avail­able for any server system in which service times are modeled by chains. The requirement that the process be regenerative is easily dropped.

Uniformization

Analysis of the transient behavior of a large or infinite Markov chain l(t) in con­tinuous time can be formidable. When the set of exit rates from all states is bounded, the process may be viewed as a Markov chain in discrete time whose transition epochs k occur at any fixed Poisson rate n exceeding the largest exit rate. This insight was first made by A. Jensen and had been ignored. The tool of uniformization was rediscovered by Keilson and Wishart [26] and its theoretical power demonstrated by Keilson in a series of papers [28,55,56]. The algorithmic importance of this tool for the study of large chains was promoted subsequently by Grassman and many other researchers.

Xl

Dynamics of Redundant Systems of Independent Repairable Components: Sojourn TImes and Exit TImes for Subsets of Large Chains

The study of the reliability of redundant systems of independent repairable com­ponents is quite difficult, even when the components have independent exponen­tially distributed failure times and repair times. The system is then described by a large Markov chain in continuous time. If a subset of the state space is specified as working states, the sojourn times on this working set and the ergodic exit time from this set are of interest. A theory of such systems was developed and a variety of simple tools for the description of such systems presented [55,56]. Many of the basic ideas useful for the description of more general large chains were set forth in a monograph (see Monograph 2 at the end of the reference list).

Retrial Systems

In retrial systems, a customer pool forms when no waiting room is available in a buffer. Each such customer then calls in at Poisson epochs until entry into the system is attained. An early contribution to retrial theory was made with J. Cozzolino and H. Young [32]. A second work with L. Servi [115] addressed the dynamics of a 2 x 2 matrix analogue of M/M/oo and provided a unified framework for some earlier retrial results.

A Circulatory Model for Human Metabolism

The study of human metabolism has been conducted by injection of low-level radioactively tagged metabolites, e.g., glucose, into the bloodstream and examina­tion of the disappearance curves for that metabolite and for transformed metabolic products, e.g., alanine. The analysis is conducted via compartmental analysis, which assumes the existence of pools of metabolites that equilibrate instantaneously. The analysis is then equivalent to that for a Markov chain with an external absorbing state, pools corresponding to states of the chain. Such pools can be spurious when the equilibration time of one or more pools is long compared to the exit rates from that pool. A more appropriate model mirrors the structure of the circulatory system as the logistical system for metabolic change. The resultant model is not that of a chain but of a semi-Markov process. A short paper [67] describing the circulatory model appeared, and a longer monograph [64] with A. Kester was issued. Unfortu­nately, the audience of medical researchers able to understand the model was too small to permit pUblication.

Monotone Matrices and Monotone Markov Processes

Monotone processes were introduced by D. Daley. The stochastic monotonicity of a Markov chain l(t) in continuous time with p(t) = exp [Qt] is latent in its infin­itesmal generator Q. For chains in discrete time l(k) with p(k) = ak, the property lies in the stochastic matrix a. A simple matrix characterization of the stochastic monotonicity of a was given with A. Kester [63] in the form t-1 a t ~ 0, where t-1

xii

and t are simple bidiagonal matrices for differencing and summation. This opera­tional characterization of stochastic monotonicity provides a useful theoretical tool. With uniformization, for example, it provides a quick proof that all birth-death processes are monotone. It also shows that monotone processes always have a second maximal eigenvalue that is positive. The same tools permit a demonstration of the unimodality of transition probabilities for birth-death processes [65].

Complete Monotonicity in Server Systems

Many of the basic descriptive distributions of interest to queueing theory are avail­able only as Laplace transforms or as infinite series of summands involving multi­ple convolution, and any simple underlying structure is concealed. The busy period distribution and waiting time distribution for MIGII are examples. The arrival stream to the server can usefully be modeled in some situations as a superposition of independent Poisson streams with exponentially distributed service times, and this is equivalent to a single Poisson stream with completely monotone p.d.f. As shown in [57], this complete monotonicity then results in the busy period being com­pletely monotone, with the waiting time distribution for customers that must wait also completely monotone. The distributions can then be described as having an exponential spectrum. It would be of value if the spectral distributions could be exploited analytically, algorithmically, or both.

The Laguerre Transform

As noted above, many of the real-domain expressions of interest in applied proba­bility, such as the busy-period density SB(t) for MIGll, require infinite sums of mul­tiple convolutions perhaps multiplied by some power t. Similar expressions, equally foreign to the natural lattice domain of computers, arise in statistical inference. The Laguerre transform method is an algorithmic procedure for mapping continuum operations into lattice operations and finally mapping the desired answer back to the continuum. The method employs Laguerre functions to build an orthonormal basis for the full continuum, and the method extends easily to higher dimensions. An extensive series of papers on this method, demonstrating its power, efficiency, accuracy, and versatility, were produced with W. Nunn and U. Sumita [73,81,85,93] and subsequently by U. Sumita and his associates.

Row Continuous Markov Chains

The state space for many important Markov chains in continuous time is a two­dimensional lattice S = {(m, n): 1 ~ m ~ M, 0 ~ n ~ N}. Then the chain l(t) may be represented as a bivariate Markov chain [M(t),N(t)]. When the row process N(t) is lattice continuous, i.e., when changes in n are 0, 1, or -1, the probability vector pTn(t) can be treated as an entity and related easily to the vectors of adjacent rows. The ergodic distribution for such a chain can then be obtained via the use of M x M matrices with greater simplicity and precision than that obtained by using matri-

xiii

ces whose order is the size of the state space. A paper describing this method and its advantages was written with U. Sumita and M. Zachmann [101].

Matrix-Geometric Distributions and Extensions

A subset of the two-dimensional chains described are row homogeneous in the sense that away from the levels near n = 0, the relation between pTn_1(t), pTn(t), and pTn+l (t) is independent of n. Chains with Markov-modulated arrival rate can be treated in this framework. The use of Green's function methods then produces easily the matrix-geometric distribution results of M. Neuts. It also shows that, for the same setting with a second boundary associated with finite system capacity, the results of Neuts extend to a second matrix-geometric decay associated with the second boundary. These results were joint with U. Sumita and/or M. Zachmann [101,107].

Uniform Stochastic Order

An influential paper with U. Sumita [80] extended the notion of stochastic order between distributions to a richer setting.

Depletion TIme for M/G/l-Type Systems

The depletion time of a single server system is the time until the server again becomes idle. The ergodic depletion time is the distribution of the depletion time from the steady state. In systems with multiple classes of customers and complex server allocation protocols with no work ever lost, the ergodic depletion time pro­vides a tight upper stochastic bound for the delay of any customer, i.e., the lowest­priority jobs. It is interesting that such customers experience a (1 - rt2 factor in their delay over high-priority customers. For congested systems with r near 1, a limit theorem is available quite different from that for the heavy traffic theorem of Kingman. A variety of results are displayed in a joint paper with U. Sumita [88].

Professor Keilson has been energetic and productive, and the above summary does not quite catch up with his recent accomplishments. Always with a strong algo­rithmic orientation in his mind for the practical use of his results, he has played an essential role in the advancement of applied probability and stochastic processes. Both academic researchers and industrial practitioners owe him a great deal. Many researchers who benefited from the works of Professor Keilson have contributed to this volume. With these authors, we are pleased to dedicate this volume to Professor Keilson in his honor.

LIST OF PUBLICATIONS BY JULIAN KEILSON

1. On Brownian motion, Boltzmann's equation, and the Fokker-Planck equation (with J. Storer). Quarterly of Applied Mathematics 10 (3), October 1952.

2. A suggested modification of noise theory. Quarterly of Applied Mathematics 12 (1), April 1954.

3. On diffusion in an external field and the adjoint source problem. Quarterly of Applied Mathematics 12 (4), January 1955.

4. The second order distribution of integrated shot noise (with N. Mermin). IRE Transactions on Information Theory, June 1959.

5. A theorem on cross-correlation between noisy channels (with P. Bello and N. Mermin). IRE Transactions on Information Theory, September 1959.

6. On time dependent queuing processes (with A. Kooharian). Annals of Mathematical Statistics 31 (1), March 1960.

7. The homogenous random walk on the half-line and the Hilbert problem. 33rd Session of L'Institut International de Statistique, Paris, France, September 1961.

8. On the general time dependent queue with a single server (with A. Kooharian). Annals of Mathematical Statistics, June 1962.

9. The use of Green's functions in the study of bounded random walks with appli­cation to queuing theory. Journal of Mathematics and Physics, March 1962.

10. Non-stationary Markov walks on the lattice. Journal of Mathematics and Physics, September 1962.

11. The general bulk queue as a Hilbert problem. Journal of the Royal Statistical Society, Ser. B, December 1962.

12. A simple random walk and an associated asymptotic behavior of Bessel func­tions. Proceedings of the Cambridge Philosophical Society (GB) 58 (4), 1962.

13. Queues subject to service interruption. Annals of Mathematical Statistics 33 (4), December 1962.

14. A Gambler's Ruin type problem in queuing theory. Journal of Operations Research 11 (4), July-August 1963.

15. The first passage time density for homogeneous skip-free walks on the continuum. Annals of Mathematical Statistics 34 (3), 375-380, September 1963.

xvi

16. On the asymptotic behavior of queues. Journal of the Royal Statistical Society, December 1963.

17. Some comments on single server queuing methods and some new results. Proceedings of the Cambridge Philosophical Society (GB),237-251,April1964.

18. An alternative to Wiener-Hopf methods for the study of bounded processes. Journal of Applied Probability 1 (1), June 1964.

19. On the ruin problem for generalized random walk. Journal of Operations Research, Letter to the Editor, June 1964.

20. The role of Green's functions in congestion theory. Symposium on Congestion Theory, University of North Carolina Press, Chapel Hill, NC, 1965.

21. A central limit theorem for processes defined on a finite Markov chain (with D. M. G. Wishart). Proceedings of the Cambridge Philosophical Society ( G B) 60,657-667,1964.

22. A review of transient behavior in regular diffusion and birth-death processes. Journal of Applied Probability 1,247-266, December 1964.

23. Boundary problems for additive processes defined on a finite Markov chain (with D. M. G. Wishart). Proceedings of the Cambridge Philosophical Society (GB) 61,173-190,1965.

24. A review of transient behavior in regular diffusion and birth-death processes - Part II. Journal of Applied Probability 2,405-428, 1965.

25. The ergodic queue length distribution for queuing systems with finite capac­ity.Journal of the Royal Statistical Society 28 (1), 190-201, 1966.

26. Addenda to processes on a finite Markov chain (with D.M.G. Wishart). Proceedings of the Cambridge Philosophical Society 63, 187-193, 1967.

27. A Limit Theorem for Passage TImes in Ergodic Regenerative Processes. Annals of Mathematical Statistics 37 (4), 866-870, 1966.

28. A technique for discussion the passage time distribution for stable systems. Journal of the Royal Statistical Society 28,477-486,1966.

29. On global extrema for a class of symmetric functions. Journal of Mathematical Analysis and Applications 18 (2),218-228,1967.

30. A theorem on optimum allocation for a class of symmetric multilinear return functions. Journal of Mathematical Analysis and Applications 15, August 1966.

31. The calculation of system failure time distributions from ergodic behavior. Rochester Society for Quality Control Transactions, 24th Annual Quality Conference, Rochester, NY, 1968.

32. A service system with unfilled requests repeated (with 1. Cozzolino and H. Young). Operations Research 16 (6), November-December 1968.

33. A note on the waiting-time distribution for the MIGll queue with last-come-first served discipline. Operations Research 16 (6), November-December 1968.

34. A queue model for interrupted communication. Opsearch 6 (1), 1969. 35. An intermittent channel with finite storage. Opsearch 6 (2), 1969. 36. On the matrix renewal function for Markov renewal processes. Annals of

Mathematical Statistics 40, 1901-1907, 1969. 37. A process with chain dependent growth rate (with S. Subba Rao). Journal of

Applied Probability 7,699-711,1970. 38. A process with chain dependent growth rate - Part II, The ruin and ergodic

xvii

problem (with S. Subba Rao). Advances in Applied Probability 3,315-338, 1971.

39. A simple non-linear integer programming algorithm for contract acceptance. Opsearch 7, 157-166, 1970.

40. Corl cycle activity in man (with C. Waterhouse). Journal of Clinical Investiga­tion 48, 2359-2366, 1969.

41. Some results for discrete unimodality (with H. Gerber). Journal of the American Statistical Association 66,386-390,1971.

42. On log-concavity and log-convexity in passage-time densities of diffusion and birth-death processes. Journal of Applied Probability 8, 391-398,1971.

43. Conversion of Convergence in the Mean to Almost Sure Convergence by Smoothing (with w. J. Hall). Center for System Science, w.P. 70-13, University of Rochester, Rochester, NY, 1970.

44. A note on summability of the entropy series. Journal of Information and Control 18, 257-260, 1971.

45. Families of infinitely divisible distributions closed under mixing and convolu­tion (with F.w. Steutel). Annals of Mathematical Statistics 43 (1),242-250,1972.

46. Markov chains and M-matrices: inequalities and equalities (with G. P. H. Styan). Journal of Mathematical Analysis and Applications 2, 439-459,1973.

47. A threshold for log-concavity. Annals of Mathematical Statistics 43, 1702-1708, 1972.

48. Canonical Factorization of Time-Series Spectra and Self-Orthogonal Series. Center for System Science, W.P. 71-01, University of Rochester, Rochester, NY, 1971.

49. On the Structure of Covariance Functions and Spectral Density Functions for Processes Reversible in Time. Center for System Science, W.P. 71-03, University of Rochester, Rochester, NY, 1971.

50. On exponential ergodicity and spectral structure for birth-death processes (with H. Callaert). Stochastic Processes and Their Applications I (2,3), 187-236, 1972.

51. Transfer times across the human body (with C. Waterhouse). Bulletin of Mathematical Biophysics 34, 33-44, 1972.

52. Mixtures of distributions, moment inequalities and measures of exponential­ity and normality (with F. W. Steutel). Annals of Probability 2 (1), 112-130, 1974.

53. Simple expressions for the mean and variance of a generalized central limit theorem and normality. Proceedings-XX International Meeting, The Institute of Management Science, 11, 538-541, 1973.

54. Compensation measures in the theory of Markov chains (with R. Syski). Stochastic Processes and Their Applications 2, 59-72, 1974.

55. Sojourn times, exit times, and jitter in multivariate Markov processes. Advances in Applied Probability 6, 747-756,1974.

56. Sojourn times, fluctuation, and rare events: the role of jitter. Advances in Applied Probability 6, 188-259, 1974.

57. Convexity and complete monotonicity in queuing distributions and associated limit behavior. Mathematical Methods in Queuing Theory, Proceedings of a

xviii

Conference at Western Michigan University, May 10--12,1974. Springer-Verlag, New York, 1974.

58. Stochastic Ordering and Renewal Theory. Department of Statistics, Stanford University, Stanford, CA, 1974.

59. Monotonicity and convexity in system survival functions and metabolic disappearance curves. Proceedings of the Conference on Reliability and Biometry, Florida State University, July 1973. SIAM, Philadelphia, 1974, pp. 81-98.

60. Passage time distributions for Gaussian Markov (Omstein-Uhlenbeck) sta­tistical processes (with H. F. Ross). Selected Tables in Mathematical Statistics (publication sponsored by the Institute of Mathematical Statistics 3,233-327, 1975.

61. Systems of independent Markov components and their transient behavior. Reliability and Fault Tree Analysis. SIAM, Philadelphia, 1975, pp. 351-364.

62. A Family of Monotone Measures of Ergodicity for Markov Chains (with 0. Vasicek). Center for System Science, w.P. 75-03, University of Rochester, Rochester, NY, October 1975.

63. Monotone matrices and monotone Markov processes (with A. Kester). Stochastic Processes and Their Applications 5 (3),231-241,1977.

64. A Circulatory Model for Human Metabolism (with A. Kester). Working Paper Series No. 7724, Graduate School of Management, University of Rochester, Rochester, NY, June 1977.

65. Unimodality preservation in Markov chains (with A. Kester). Stochastic Processes and Their Applications 7 (2) 179-190,1978.

66. Exponential spectra as a tool for the study of server-systems with several classes of customers. Journal of Applied Probability 15, 162-170, 1978.

67. A circulatory model for human metabolism (with A. Kester and C. Waterhouse). Journal of Theoretical Biology 74, 535-547, 1978.

68. Oscillating Brownian motion (with 1. Wellner). Journal of Applied Probability 15,300--310,1978.

69. The contribution of glucose to alanine metabolism in man (with C. Water­house). Journal of Laboratory and Clinical Medicine 92, 803-812, November 1978.

70. Possible impact of the new spectrometric techniques on 14/C tracer kinetic studies (with C. Waterhouse). In: Gove, H. E. (ed), Proceedings of the Confer­ence on Radio-Carbon Dating with Accelerators. University of Rochester, Rochester, NY, April 20--21, 1978.

71. Logistic failure vs mission failure in reliability specification (with S. C. Graves). Seventh Annual Acquisition Research Symposium, Department of the Air Force, Hershey, PA, May 31-June 2, 1978.

72. Some Dangers in the Use of Exponential Distribution Assumptions in Stochas­tic Models. Working Paper Series No. 7810, Graduate School of Management, University of Rochester, Rochester, NY, June 1978.

73. Laguerre transformation as a tool for the numerical solution of integral equa­tions of convolution type (with W. Nunn). Applied Mathematics and Compu­tation 5,313-359, 1979.

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74. The Maximum of the Stationary Gaussian Markov Process over an Interval -Theory, Table and Graphs (with H. Ross), August 1978.

75. The maximum over an interval of meterological variates modeled by the sta­tionary Gaussian Markov processes (with H. Ross). Preprint Volume of Sixth Conference on Probability and Statistics in Atmospheric Sciences, October 9-12,1979,pp.213-216.

76. On the unimodality of passage time densities in birth-death processes. Statis­tica Neederlandica 35 (1),49-55,1981.

77. A methodology for studying the dynamics of extended logistic systems (with Stephen C. Graves). Naval Research Logistics Quarterly 26 (2), 169-197, June 1979.

78. The compensation method applied to a one-product/inventory problem (with S. C. Graves). Mathematics of Operations Research 6 (2),246-262, May 1981.

79. Extrapolation of the mean lifetime of a large population from its preliminary survival history (with U. Sumita). Naval Research Logistics Quarterly 30, 509-535,1983.

80. Uniform stochastic ordering and related inequalities (with U. Sumita). Canadian Journal of Statistics 10, 181-198, 1982.

81. The bilateral Laguerre transform (with W. Nunn and U. Sumita). Applied Mathematics and Computation 8 (2), 137-174, March 1981.

82. On the distribution and covariance structure of the present value of a random income stream. Journal of Applied Probability 19,240-244, 1982.

83. A decomposition of the beta distribution, related order and asymptotic behav­ior (with U. Sumita). Annals of the Institute of Statistics and Mathematics A, 35 (2),243-253,1983.

84. Significance points for some tests of uniformity on the sphere (with D. Petron­das, U. Sumita, and J. Wellner). Journal of Statistical Computation and Simula­tion 17 (2), 195-218, 1983.

85. Waiting time distribution response to traffic surges via the Laguerre transform (with U. Sumita). Applied Probability - Computer Science: The Interface, Vol. II. Birkauser, 1982, pp. 109-133.

86. System balance for extended logistic systems (with S. C. Graves). Operations Research 31 (2),234-252,1983.

87. The Ehrenfest Chain as an Approximation to the O-U Process (with U. Sumita and M. Zachmann). Working Paper Series No. 8119, Graduate School of Management, University of Rochester, Rochester, NY, July 1981.

88. The depletion time for M/G/1 systems and a related limit theorem (with U. Sumita). Advances in Applied Probability 15,420-443, 1983.

89. Parts and service demand distribution generated by primary production (with P. Kubat). European Journal of Operations Research 17,257-265, 1984.

90. Evaluation of the total time in system in a preempt resume priority queue via a modified Lindley process (with U. Sumita). Advances in Applied Probability 15,840-856,1984.

91. Stochastic systems. Course notes, Massachusetts Institute of Technology, pp. 1-155, Spring 1962.

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92. Convergence of quasi-stationary distributions in birth-death processes (with R. Ramaswamy). Stochastic Processes and Their Applications 18, 301-312, 1984.

93. A general Laguerre transform and a related distance between probability mea­sures (with U. Sumita). Journal of Mathematical Analysis and Applications 113 (1),288-308, January 1986.

94. A Time-Dependent Model of a Two Echelon System with Quick Repair (with U. Sumita). Working Paper Series OM 8311, Graduate School of Management, University of Rochester, Rochester, NY, May 1983.

95. Oscillating random walk models for GI/G/l vacation systems with Bernoulli schedules (with L. Servi).Journal of Applied Probability 23, 790-802, 1986.

96. The bivariate maximum process and quasi-stationary structure (with R. Ramaswamy). Stochastic Processes and Their Applications, 27-36,1986.

97. The relaxation time of single server queuing systems with Poisson arrivals and hyper-exponential/Erlang service times (with F. Machihara and U. Sumita). Proceedings of the 11th International Teletraffic Congress, Kyoto, Japan, 3, 242-251, September 1985.

98. Hyper-exponential waiting time structure in hyper-exponential G/G/1 systems (with F. Machihara). Journal of the Operations Research Society of Japan 28 (3),242-250,1985.

99. Stochastic models in reliability theory. Theory of Reliability, International School of Physics Enrico Fermi Course XCIV, North Holland Press, Amster­dam, February 1986, pp. 23-53.

100. The backlog and depletion time process for M/G/l vacation models with exhaustive service discipline (with R. Ramaswamy). Journal of Applied Prob­ability, 404-412,1986.

101. Row-continuous finite Markov chains: structure and algorithms (with U. Sumita and M. Zachmann). Journal of the Operations Research Society of Japan 3,291-314,1987.

102. Return of the wanderer: a physicist becomes a probabilist. In: Gani, J. (ed), Julian Keilson, The Craft of Probabilistic Modeling - A Collection of Personal Accounts. Springer-Verlag New York, 1986, pp. 166-185.

103. Robustness and exponentiality in redundant repairable systems. Annals of Operations Research 9,439-447,1987.

104. The dynamics of the M/G/l vacation model (with L. Servi). Operations Research 35 (4),575-582,1987.

105. The relaxation time for truncated birth-death processes (with R. Ramaswamy). Probability in the Engineering and Information Sciences 1, 367-381,1987.

106. MlG/co with batch arrivals (with A. Seidmann). Operations Research Letters 7, 219-222, 1988.

107. Homogeneous row-continuous bivariate Markov chains with boundaries (with M. Zachmann). Journal of Applied Probability, 25A (A Celebration of Applied Probability), 237-256,1988.

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108. A distributional form of Little's law (with L. Servi). Operations Research Letters 7,223-227,1988.

109. A clocked schedule with feedback (with L. Servi). Performance Evaluation 9, 191-208,1989.

110. Blocking probability for M/G/1 vacation systems with occupancy level dependent schedules (with L. Servi). Operations Research 37, 134-140, 1989.

111. A distributed Poisson approximation for preempt-resume clocked schedules (with L. Servi). IEEE Transactions on Communications 38 (2), 192-198, 1990.

112. The distributional form of Little's law and the Fuhrmann-Cooper decompo­sition (with L. Servi). Operations Research Letters, 239-247,1990.

113. Transient and busy period analysis of the G/G/1 queue as a Hilbert problem. Journal of Applied Probability 28, 873-885, 1991.

114. Spectral structure of M/G/1 systems - asymptotic behavior and relaxation time (with F. Machihara and U. Sumita). The Institute of Electronics, Informa­tion and Communication Engineers, Transactions on Communications E7S-B (12),1992.

115. The matrix M/M/oo system: retrial models and Markov modulated sources (with L. Servi). Journal of Applied Probability, •• 1993.

116. Networks of non-homogeneous M/G/oo systems (with L. D. Servi). Journal of Applied Probability, 157-168, 1994.

117. The M/G/1IK blocking formula and its generalizations (with L. Servi). Queuing Systems: Theory and Applications, 1993.

118. ATM cell multiplexer analysis by a compensation method (with A E. Conway). IEEE Transactions on Communications 42 (2-4), 1994.

119. Delay distribution in a two-stage finite buffer queuing model with blocking and flow control (with AE. Conway). Working paper, GTE Labs, 1992. Submitted to Performance Evaluation.

120. Threshold queues (with 0. Ibe). Working paper, GTE Labs, 1992. Submitted to European Journal of Operational Research.

121. Multi-server threshold queues with hysteresis (with O. Ibe). Performance Eval­uation 21, 185-213, 1995.

122. Heterogeneous multiserver threshold queues with hysteresis (with 0. Ibe). Working paper, GTE Labs, 1992. Submitted to Performance Evaluation.

123. Cutoff priority scheduling in mobile cellular communication systems (with 0. Ibe). IEEE Transactions on Communications 43, 1038-1045, 1995.

124. Overload control in finite-buffer multiclass message systems (with 0. Ibe). Telecommunication Systems 2, 121-140, 1994.

125. Dynamic bandwidth, allocation, routing and access control in ATM networks (with A Gersht, A Shulman, and 1. Vucetic). To appear in: Frisch, I., Malek, M., and Panwar, S. (eds), Network Management and Control, Vol. 2.

Monograph 1. Green's Function Methods in Probability Theory. Charles Griffin and Company, Ltd., 1965.

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Monograph 2. Markov Chain Models - Rarity and Exponentiality. Springer­Verlag, Applied Mathematical Sciences Series 28, 1979.

Selected Tables in Mathematical Statistics, Vol. 3, 1975, Institute of Mathematical Statistics: Keilson, 1., and Ross, H., Passage time distributions for Gaussian-Markov (Ornstein-Uhlenbeck) statistical processes, pp. 233-327.